On the maximum of randomly weighted sums with regularly varying tails

Consider the randomly weighted sums Sn(θ)=∑k=1nθkXk, n=1,2,…, where {Xk,k=1,2,…} is a sequence of independent real-valued random variables with common distribution F, whose right tail is regularly varying with exponent -α<0, and {θk,k=1,2,…} is a sequence of positive random variables, independent of {Xk,k=1,2,…}. Under a suitable summability condition on the upper endpoints of θk,k=1,2,…, we prove that Pr(max1⩽n<∞Sn(θ)>x)∼F¯(x)∑k=1∞Eθkα.